Method for estimating T2

ABSTRACT

A method for providing an estimated 3D T 2  map for magnetic resonance imaging using a Double-Echo Steady-State (DESS) sequence for a volume of an object in a magnetic resonance imaging (MRI) system is provided. A DESS scan of the volume is provided by the MRI system. Signals S 1  and S 2  are acquired by the MRI system. Signals S 1  and S 2  are used to provide a T 2  map for a plurality of slices of the volume, comprising determining repetition time (TR), echo time (TE), flip angle α, and an estimate of the longitudinal relaxation time (T 1 ), and wherein the DESS scan has a spoiler gradient with an amplitude G and a duration τ and ignoring echo pathways having spent more than two repetition times in the transverse plane.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority under 35 U.S.C. § 119(e) from U.S.Provisional Application No. 62/324,328, entitled “SIMPLE ESTIMATION OFT2 MAPS USING THE DOUBLE-ECHO STEADY-STATE SEQUENCE”, filed Apr. 18,2016.

GOVERNMENT RIGHTS

This invention was made with Government support under contracts AR063643and EB002524 and P41EB015891 awarded by the National Institutes ofHealth. The Government has certain rights in the invention.

BACKGROUND OF THE INVENTION

This invention relates generally to magnetic resonance imaging (MRI).Magnetic resonance imaging (MRI) is a non-destructive method for theanalysis of materials and is an approach to medical imaging. It isgenerally non-invasive and does not involve ionizing radiation. In verygeneral terms, nuclear magnetic moments are excited at specific spinprecession frequencies which are proportional to the local magneticfield. The radio-frequency signals resulting from the precession ofthese spins are received using pickup coils. By manipulating themagnetic fields, an array of signals is provided representing differentregions of the volume. These are combined to produce a volumetric imageof the nuclear spin density of the body.

Magnetic resonance (MR) imaging is based on nuclear spins, which can beviewed as vectors in a three-dimensional space. During MRI, each nuclearspin responds to four different effects: precession about the mainmagnetic field, nutation about an axis perpendicular to the main field,and both transverse and longitudinal relaxation.

SUMMARY OF THE INVENTION

In accordance with the invention, a method for providing an estimated 3DT₂ map for magnetic resonance imaging using a Double-Echo Steady-State(DESS) sequence for a volume of an object in a magnetic resonanceimaging (MRI) system is provided. A DESS scan of the volume is providedby the MRI system. Signals S₁ and S₂ are acquired by the MRI system.Signals S₁ and S₂ are used to provide a T₂ map for a plurality of slicesof the volume, comprising determining repetition time (TR), echo time(TE), flip angle α, and an estimate of the longitudinal relaxation time(T₁), and wherein the DESS scan has a spoiler gradient with an amplitudeG and a duration τ and ignoring echo pathways having spent more than tworepetition times in the transverse plane.

In another manifestation, a method for providing an estimated 3D T₂ mapfor magnetic resonance imaging using a Double-Echo Steady-State (DESS)sequence for a volume of an object in a magnetic resonance imaging (MRI)system is provided. A DESS scan of the volume is provided by the MRIsystem. Signals S₁ and S₂ are acquired by the MRI system. S₁ and S₂ areused to provide a T₂ map for a plurality of slices of the volume, usingthe equation

${T_{2} = \frac{{- 2}\left( {{TR} - {TE}} \right)}{\log\left\lbrack {\left( \frac{S_{2}}{S_{1}\sin^{2}\frac{\alpha}{2}} \right){f\left( {\alpha,{TR},{T\; 1}} \right)}} \right\rbrack}},$wherein ƒ(α, TR, T1) is a function that is not proportional to

$\sin^{2}\frac{\alpha}{2}$and TR is a repetition time, TE is an echo time, and α is a flip angle.

The invention and objects and features thereof will be more readilyapparent from the following detailed description and appended claimswhen taken with the drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed incolor. Copies of this patent or patent application publication withcolor drawing(s) will be provided by the Office upon request and paymentof the necessary fee.

FIG. 1 is a series of graphic displays of an embodiment.

FIG. 2 is a series of graphs of signals of an embodiment.

FIG. 3 illustrates results from a phantom scan.

FIG. 4 is a series of images that show T₂ estimates from a DESS scan.

FIG. 5 is a series of images that show the scan of a patient with ahistory of knee injuries and osteochondral defects.

FIG. 6 shows sensitivity of the T₂ estimate to errors in the T₁ and aassumptions.

FIG. 7 is a high level flow chart of an embodiment of the invention.

FIG. 8 is a schematic top view of a magnetic resonance imaging (MRI)system that may be used in an embodiment of the invention.

FIG. 9 illustrates a computer system that may be used in an embodimentof the invention.

DETAILED DESCRIPTION OF ILLUSTRATED EMBODIMENTS

1. Introduction

The Double-Echo Steady-State (DESS) sequence offers distortion-free,SNR-efficient 3D-imaging with good contrast flexibility. DESS is agradient-spoiled steady-state sequence, collecting two echoes perrepetition (TR), with the spoiler gradient separating the two echoes.The two signals, often labeled S₁ (before the spoiler) and S₂ (after thespoiler), have different contrast. The S₁ signal is often said to haveT₁/T₂ weighting and S₂ to have a mixture of T₂ and diffusion weighting,but in reality both the contrasts are quite complicated functions of theextrinsic scan parameters (TR, TE, flip angle α, gradient amplitude Gand gradient duration τ) as well as the intrinsic tissue parameters (T₁,T₂, and diffusivity). While DESS has been used for various anatomies,such as the breast and the prostate, it has found particularlywidespread use for musculoskeletal imaging, where, for example, it hasbeen used in the Osteoarthritis Initiative (OAI) to determineOsteoarthritis (OA) progression.

To improve OA detection, DESS has been used for quantitative imaging ofT₂ and apparent diffusion coefficient (ADC). These have been found tocorrelate with collagen network organization and proteoglycanconcentration, respectively. A number of signal models have beendeveloped to quantitatively describe both of the DESS signals and theirdependency on tissue parameters. These include both closed-form andnon-closed-form analytical methods as well as numerical methods.

Some methods model the DESS signals completely and can be consideredfully accurate for a single T₁, T₂, and diffusivity in a voxel. However,owing to their complexity, using these models for parameter estimationtypically requires solving them with a numerical search over apredetermined solution set. This inhibits immediate, automatic creationof parameter maps right at the scanner, which would be of great valuefor clinical scans. Furthermore, a method in Freed D E, Scheven U M,Zielinski L J, Sen P N, Hürlimann M D. Steady-state free precessionexperiments and exact treatment of diffusion in a uniform gradient.Journal of Chemical Physics. 2001; 115(9):4249-4258, which isincorporated by reference for all purposes, and which describes themagnetization with a continued fraction, does not allow for pulsedgradients. For these reasons, approximate models are often used toestimate the desired parameter in spite of known errors. For example, inWelsch G H, Scheffler K, Mamisch T C, Hughes T, Millington S, DeimlingM, Trattnig S. Rapid Estimation of Cartilage T₂ Based on Double Echo atSteady State (DESS) With 3 Tesla. Magnetic Resonance in Medicine. 2009;62:544-549, which is incorporated by reference for all purposes, thefollowing model, originally suggested in Bruder H, Fischer H, GraumannR, Deimling M. A New Steady-State Imaging Sequence for SimultaneousAcquisition of Two MR Images with Clearly Different Contrasts. MagneticResonance in Medicine. 1988; 7:35-42, which is incorporated by referencefor all purposes, was used to estimate T₂ from a single DESS scan:

$\begin{matrix}{\frac{S_{2}}{S_{1}} = e^{- \frac{2{({{TR} - {TE}})}}{T\; 2}}} & (1)\end{matrix}$

This relationship essentially describes a spin echo relationship betweenS₂ and S₁ in consecutive TRs. Although this approach may offer usefulresults in spite of errors in T₂, practical T₂ mapping would benefitfrom a comparably simple analytical formula that is even more accurate.

An embodiment of the invention provides a model that can be used for T₂estimation from a single DESS scan. Our analysis demonstrates thathigher-order echo pathways can often be ignored, based on pathwaycancellation effects and signal decay, leading to a simple and accuratelinear relationship between the signals from two DESS echoes. Wevalidate the overall approach using simulations, phantom scans, and invivo knee scans.

2. Theory

2.1 Derivation of Simplified Signal Ratio

A simple yet accurate linear approximation of the relationship betweenthe two DESS echoes can be derived by tracing the components thatcontribute to the magnetization, starting at S₂, until we arrive at S₁.We use the Extended Phase Graph (EPG) formalism, which decomposesmagnetization into a basis of positively and negatively dephasedtransverse states (coefficients F₀, F₁, F₂, . . . ) as well assinusoidally varying longitudinal states (coefficients Z₀, Z₁, Z₂, . . .). The procedure is explained graphically in FIGS. 1a-c , using similargraphical structures as those introduced by Scheffler K. A PictorialDescription of Steady-States in Rapid Magnetic Resonance Imaging.Concepts in Magnetic Resonance. 1999; 11(5):291-304, which isincorporated by reference for all purposes. In the EPG formalism, themagnetization can be represented by a matrix of coefficients (F_(n) andZ_(n)) that multiply transverse states with n cycles of positive ornegative dephasing, and longitudinal states with n cycles of sinusoidalvariation:

$\begin{matrix}{M = \begin{bmatrix}F_{0} & F_{1} & F_{2} & \ldots \\F_{0}^{\star} & F_{- 1}^{\star} & F_{- 2}^{\star} & \ldots \\Z_{0} & Z_{1} & Z_{2} & \ldots\end{bmatrix}} & (2)\end{matrix}$where an asterisk (*) denotes a complex conjugate. The observable signalis the DC transverse term F₀. Relaxation and diffusion over time arerepresented by scaling of the elements (T₁ relaxation is represented byan additive factor to the DC longitudinal term, Z₀) and the effect of agradient is represented by shifting the transverse coefficients F_(n).

The effects of T₁/T₂ decay, as well as diffusion, on state n, can bedescribed by the scaling terms

${\overset{\sim}{E}}_{2,n} = {{e^{{- \frac{TR}{T\; 2}} - {{\lbrack{{{({n^{2} + n + \frac{1}{2}})}{TR}} - \frac{\tau}{6}}\rbrack}\Delta\; k^{2}D}}\mspace{14mu}{and}\mspace{14mu}{\overset{\sim}{E}}_{1,n}} = e^{{- \frac{TR}{T\; 1}} - {n^{2}{TR}\;\Delta\; k^{2}D}}}$(assuming the gradient in the center of the TR). The effect of an RFpulse with flip angle α and phase ϕ is represented by matrixmultiplication with the matrix:

$\begin{matrix}{{R\left( {\alpha,\phi} \right)} = \begin{bmatrix}{\cos^{2}\frac{\alpha}{2}} & {e^{2i\;\phi}\sin^{2}\frac{\alpha}{2}} & {{- {ie}^{i\;\phi}}\sin\mspace{11mu}\alpha} \\{e^{{- 2}i\;\phi}\sin^{2}\frac{\alpha}{2}} & {\cos^{2}\frac{\alpha}{2}} & {{ie}^{{- i}\;\phi}\sin\mspace{11mu} a} \\{{- \frac{i}{2}}e^{{- i}\;\phi}\sin\mspace{11mu}\alpha} & {\frac{i}{2}e^{i\;\phi}\sin\mspace{11mu}\alpha} & {\cos\mspace{11mu}\alpha}\end{bmatrix}} & (3)\end{matrix}$

Here, TR and TE represent the repetition and echo time, respectively, αis the flip angle, D is the diffusivity, and the dephasing per unitlength induced by the unbalanced gradient is denoted by Δk=γGτ, where Gand τ are the spoiler amplitude and duration, respectively, and γ is thegyromagnetic ratio. We label the states immediately before/after the RFpulse with −/+ superscripts, respectively. To determine a relationshipbetween the two DESS signals, we look at the echo pathway of theobservable signal immediately before the pulse, F₀ ⁻, and write down thecomponents that contributed to it. We repeat this until we reach theobservable signal immediately after the pulse, F₀ ⁺ (FIG. 1). Asjustified in Section 2.2, we will ignore signal contributions fromstates that have experienced two cycles of dephasing or more, as wasdone for the S₂ signal in Buxton R B. The Diffusion Sensitivity of FastSteady-State Free Precession Imaging. Magnetic Resonance in Medicine.1993; 29:235-243, which is incorporated by reference for all purposes.We will assume ϕ=90°, resulting in real valued states, without loss ofgenerality.

$\begin{matrix}\begin{matrix}{F_{0}^{-} = {{{\overset{\sim}{E}}_{2,{- 1}}F_{- 1}^{+}} = {{\overset{\sim}{E}}_{2,{- 1}}\left( {{{- \sin^{2}}\frac{\alpha}{2}F_{1}^{-}} + {\cos^{2}\frac{\alpha}{2}F_{- 1}^{-}} - {\sin\mspace{11mu}\alpha\mspace{11mu} Z_{1}^{-}}} \right)}}} \\{= {- {{\overset{\sim}{E}}_{2,1}\left( {{\sin^{2}\frac{\alpha}{2}{\overset{\sim}{E}}_{2,0}F_{0}^{+}} - {\cos^{2}\frac{\alpha}{2}{\overset{\sim}{E}}_{2,{- 2}}F_{- 2}^{+}} + {\sin\mspace{11mu}\alpha\;{\overset{\sim}{\; E}}_{1,1}Z_{1}^{+}}} \right)}}} \\{\approx {- {{\overset{\sim}{E}}_{2,{- 1}}\left( {{\sin^{2}\frac{\alpha}{2}{\overset{\sim}{E}}_{2,0}F_{0}^{+}} + {\sin\mspace{11mu}\alpha\mspace{11mu}{\overset{\sim}{E}}_{1,1}Z_{1}^{+}}} \right)}}}\end{matrix} & \left( {4a} \right) \\\begin{matrix}{Z_{1}^{+} = {{\frac{1}{2}\sin\mspace{11mu}\alpha\mspace{11mu} F_{1}^{-}} + {\frac{1}{2}\sin\mspace{11mu}\alpha\mspace{11mu} F_{- 1}^{-}} + {\cos\mspace{11mu}\alpha\mspace{11mu} Z_{1}^{-}}}} \\{= {{\frac{1}{2}\sin\mspace{11mu}\alpha\mspace{11mu}{\overset{\sim}{E}}_{2,0}F_{0}^{+}} + {\frac{1}{2}\sin\mspace{11mu}\alpha\mspace{11mu}{\overset{\sim}{E}}_{2,{- 2}}F_{- 2}^{-}} + {\cos\mspace{11mu}\alpha\mspace{11mu}{\overset{\sim}{E}}_{1,1}Z_{1}^{-}}}} \\{\approx {{\frac{1}{2}\sin\mspace{11mu}\alpha\mspace{11mu}{\overset{\sim}{E}}_{2,0}F_{0}^{+}} + {\cos\mspace{11mu}\alpha\mspace{11mu}{\overset{\sim}{E}}_{1,1}\left. Z_{1}^{-}\Longrightarrow Z_{1}^{+} \right.}} \approx {\frac{1}{2}\frac{\sin\mspace{11mu}\alpha\mspace{11mu}{\overset{\sim}{E}}_{2,0}}{1 - {\cos\mspace{11mu}\alpha\mspace{11mu}{\overset{\sim}{E}}_{1,1}}}F_{0}^{+}}}\end{matrix} & \left( {4b} \right)\end{matrix}$

Combining Eqs. (4a-b) and using some algebra and trigonometricidentities yields

$\begin{matrix}{\frac{F_{0}^{-}}{F_{0}^{+}} \approx {{- {\overset{\sim}{E}}_{2,{- 1}}}{\overset{\sim}{E}}_{2,0}{\sin^{2}\left( \frac{\alpha}{2} \right)}\frac{1 + {\overset{\sim}{E}}_{1,1}}{1 - {\cos\mspace{11mu}\alpha\mspace{11mu}{\overset{\sim}{E}}_{1,1}}}}} & (5)\end{matrix}$

Using the stated expressions for the decay factors as well as the factthat the observed signal magnitudes are

$S_{1} = {{F_{0}e^{- \frac{TE}{T\; 2^{\star}}}\mspace{14mu}{and}\mspace{14mu} S_{2}} = {{- {\overset{\sim}{E}}_{2,{- 1}}}F_{- 1}e^{\frac{TE}{T\; 2}}e^{{- \frac{TE}{T\; 2}},}}}$(where we have used that the two signals are out of phase and the secondsignal experiences T₂′ rephasing), we arrive at the followingexpression:

$\begin{matrix}{\frac{S_{2}}{S_{1}} = {e^{{- \frac{2{({{TR} - {TE}})}}{T\; 2}} - {{({{TR} - \frac{\tau}{3}})}\Delta\; k^{2}D}}{\sin^{2}\left( \frac{\alpha}{2} \right)}\left( \frac{1 + e^{{- \frac{TR}{T\; 1}} - {{TR}\;\Delta\; k^{2}D}}}{1 - {\cos\mspace{11mu}\alpha\mspace{11mu} e^{{- \frac{TR}{T\; 1}} - {{TR}\;\Delta\; k^{2}D}}}} \right)}} & (6)\end{matrix}$

In each panel of FIGS. 1a-c , we consider the magnetization at the timeof the right blue dot and examine the components from the time of theleft blue dot that contributed to it. Graphical representations areshown of transverse and longitudinal states (above and below the dottedline, respectively). The accrued phase of the transverse magnetizationis represented by the distance of the black line from the time axis. (a)The state that contributes to the measurable signal (S2) before the RFpulse is the “negatively dephased” signal following the previous RFpulse. (b) Two relevant paths before the RF pulse contribute to thestate in panel 1 a, shown with black arrows. These are transverse andlongitudinal states that have a net dephasing from one gradient. Theyalso contribute to the longitudinal state following the RF pulse (grayarrows). (c) The measurable signal following the previous RF pulse (S1),before dephasing by the gradient, contributes to the “positivelydephased” transverse pathway from FIG. 1b . This is the desired endpointso the pathway is not retraced further backward. The longitudinalpathway is not dephased by the gradient. The contribution of a thirdtransverse pathway has been ignored since it comes from a doublydephased signal, which can be neglected (shown with red x's). This givesa recursive relationship between S1 and S2 that can be easily solved,resulting in Eq. (6).

This relationship, which assumes the echoes are acquired symmetricallyduring the TR (with the same duration from the RF pulse to S₁ as from S₂to the subsequent RF pulse), provides a simple analytic estimate of T₂given an estimate of T₁ and either minimal diffusion sensitivity or anestimate of diffusivity.

In the case of low spoiling (Δk≈0), the model reduces to the followingrelationship:

$\begin{matrix}{\frac{S_{2}}{S_{1}} = {e^{- \frac{2{({{TR} - {TE}})}}{T\; 2}}{\sin^{2}\left( \frac{\alpha}{2} \right)}\left( \frac{1 + e^{- \frac{TR}{T\; 1}}}{1 - {\cos\mspace{11mu}\alpha\mspace{11mu} e^{- \frac{TR}{T\; 1}}}} \right)}} & (7)\end{matrix}$

For α=90°, this further reduces to

$\begin{matrix}{\frac{S_{2}}{S_{1}} = {e^{- \frac{2{({{TR} - {TE}})}}{T\; 2}}\frac{1 + e^{- \frac{TR}{T\; 1}}}{2}}} & (8)\end{matrix}$

In the case of very long T₁, Eq. (7) becomes independent of flip angleand results in Eq. (1). This equation was presented by Bruder et al. andhas been used by Welsch et al, for T₂ estimation in cartilage. However,this relationship results in errors unless the flip angle is close to90° or TR/T₁ is very small. The relationship also assumes weak spoilingso that diffusion effects may be ignored. Eq. (6) better approximatesthe signal over a large range of T₁, flip angle α, and (zeroth) spoilergradient moments. This is demonstrated in FIG. 2a , which shows theapproximated signal from Eqs. (1) and (6), as well as highly accuratenumerical computation of the signal using EPG matrices. The signal ratiois modeled with a typical T₁ value of 1.2 s for cartilage, assuming bothweak and strong spoiling (with spoiler moments of 0.001 and 156.6mT/m·ms, respectively), as well as a long T₁ value of 5 s for the weaklyspoiled case, and a TR of 22.5 ms. Other parameters were TE=6.5 ms,T₂=40 ms, gradient duration τ=3.4 ms, and ADC=1.6·10⁻³ mm/s. In orderfor Eq. (1) to be valid with smaller flip angles (about 20°-60°), T₁clearly needs to be much larger than TR.

FIG. 2a shows comparison of proposed model of Eq. (6) (dashed) withcomplete simulations (colored solids) and with the simple exponentialmodel of Eq. (1) (black solid) in a tissue with T₂=40 ms. The blue(bottom) curves correspond to strong spoiling (gradient moment 156.6mT/m·ms) and the green (middle) curves to weak spoiling (gradient moment0.001 mT/m·ms), both with T₁=1.2 s. The red (top) curves correspond toweak spoiling and T₁=5 s. The green curves represent DESS scans that canbe used for T₂ estimation in cartilage.

2.2 Cancellation of Higher-Order Echo Pathways

The model in Eq. (6) performs well for two reasons. The equationneglects the contribution of echo pathways between S₁ and S₂ that havespent more than two TRs in the transverse plane. This is partlyjustified by simple signal decay—the more time that the magnetizationspends in the transverse plane, the more it will be attenuated both byT₂ relaxation and diffusion. However, a more important mechanism is thecancellation of echo pathways due to opposing phase, independent ofrelaxation. It is easy to show that all contributions to S₂ that haveexperienced two transverse TRs will have negative phase relative to S₁.However, this does not hold for higher order echo pathways. For example,if we denote TRs where the magnetization starts in the +/−n^(th)transverse state or the n^(th) longitudinal state by F_(−/−n) and Z_(n),then the pathways F₀F₁F⁻²F⁻¹ and F₀Z₁F₁F⁻²F⁻¹ will have opposite sign.By accumulating the contributions of all such pathways, the positive andnegative contributions (relative to S₁) of higher-order pathways can bemodeled and compared. For example, by including pathways with two andfour transverse TRs, Eq. (6) can be extended to

$\begin{matrix}{\frac{S_{2}}{S_{1}} = {B_{2 -} - B_{4 +} + B_{4 -}}} & (9)\end{matrix}$where B²⁻ is the two-TR component given by Eq. (6), and B₄₊ and B⁴⁻ arethe four-TR components with the same and opposite phase to S₁,respectively, given by

$\begin{matrix}{B_{4 +} = {2{CDEF}}} & (10) \\{B_{4 -} = {{C\left( {D^{2} + E^{2}} \right)}F}} & (11) \\{where} & \; \\{C = e^{{- \frac{{4{TR}} - {2{TE}}}{T\; 2}} - {{({{6{TR}} - \frac{2\tau}{3}})}\Delta\; k^{2}D}}} & (12) \\{D = {\cos^{2}\left( \frac{\alpha}{2} \right)}} & (13) \\{E = \frac{\frac{1}{2}\sin^{2}\alpha\mspace{11mu} e^{{- \frac{TR}{T\; 1}} - {{TR}\;\Delta\; k^{2}D}}}{1 - {\cos\mspace{11mu}\alpha\mspace{11mu} e^{{- \frac{TR}{T\; 1}} - {{TR}\;\Delta\; k^{2}D}}}}} & (14) \\{F = {{\sin^{2}\left( \frac{\alpha}{2} \right)} + \frac{\frac{1}{2}\sin^{2}\alpha\mspace{11mu} e^{{- \frac{TR}{T\; 1}} - {4{TR}\;\Delta\; k^{2}D}}}{1 - {\cos\mspace{11mu}\alpha\mspace{11mu} e^{{- \frac{TR}{T\; 1}}4{TR}\;\Delta\; k^{2}D}}}}} & (15)\end{matrix}$

FIG. 2b displays the components B_(n+) and B_(n−) for n=2-12 over arange of flip angles with the same parameters as the green curve in FIG.2a . FIG. 2b shows positive (dashed) and negative (solid) contributionsto S₂/S₁ from pathways spending 2-12 TRs in the transverse plane. Theblack curves represent the sum of the different pathways. For pathwaysof order 6 and above, the positive (×) curves overlap with the negative(solid) curves. The higher-order components with positive and negativephase have very similar magnitude. Therefore, they will have anegligible effect on the final answer, particularly for flip anglesabove 20° or short T₂. This is demonstrated in FIG. 2c . This leads toEq. (6) simulating the signal ratio S₂/S₁ very accurately, even in caseswhere there is very little T₂ decay, such as in fluid. FIG. 2c shows netcontributions from the pathways in FIG. 2b shows that pathways with morethan 2 TRs in the transverse plane contribute minimally. FIG. 2d showsthe same simulation as FIG. 2c , but for synovial fluid at 3T, havingT₁=3.6 s and T₂=0.77 s, showing that the proposed model (blue signal)approximates the true signal (black curve) well for flip angles above20°, even for long T₁ and T₂. The higher-order pathways contribute moreto the signal ratio in this case, especially for low flip angles, butEq. (6) nonetheless approximates the true signal well for flip anglesabove 20°.

3. Methods

In an embodiment, all scans were performed on a 3T 750 scanner (GEHealthcare, Waukesha, Wis., USA). Informed consent was obtained from allsubjects in accordance with the Institutional Review Board protocol atour institution. The volunteer scans used a 16-channel receive coilwrapped around the knee (GEM Flex by NeoCoil, Waukesha, Wis., USA), thephantom scans used a single-channel transmit-receive wrist coil (GEBC-10 by Mayo Clinic, Rochester Minn., USA), and the patient scans usedan 8-channel transmit-receive knee coil (Knee Array Coil by Invivo,Gainesville, Fla., USA).

3.1 Phantom Scans

Three agar phantoms were scanned using DESS with scan parameters TR=13.4ms, TE=3.9 ms, voxel size 0.47×0.47×3.0 mm³, spoiler moment 7.83 mT/m·msand duration τ=3.4 ms, and flip angles α=10°, 15°, 20°, 30°, 40°, and50°. Fast spin-echo (FSE) scans with echo times TE=17, 26, 34, 43, 51,60, and 68 ms, TR=1000 ms and voxel size 0.47×0.47×6.0 mm³ wereperformed and fitted to a monoexponential curve to obtain a referenceT₂. The T₂ value was estimated from the DESS scans using Eqs. (1) and(7) and compared to the value resulting from the reference FSE scans.For Eq. (7), reference T₁ values known to be T₁₋₁=2000 ms, T₁₋₂=1700 ms,and T₁₋₃=1600 ms, measured with a 3TI MP-RAGE technique, and a flipangle measured with the Bloch-Siegert method, were used for processingeach phantom.

3.2 Volunteer Scans

To test Eq. (7) in vivo, a sagittal DESS scan was performed in the kneeof three healthy volunteers. The scans used a spoiler with a moment of15.66 mT/m·ms and duration 3.4 ms. Other parameters were α=25°, TR=22.5ms, TE=6.5 ms, slice thickness 3.0 mm. A spectrally selective RF pulsewas used. The T₂ of the cartilage was again estimated using the linearapproximations of Eqs. (1) and (7). Reference FSE scans were acquiredwith TE=9, 18, 27, 36, 45, 54, 63, and 72 ms and TR=1000 ms. Both scanswere run with FOV=13 cm×13 cm and acquisition matrix 384×384, which wasautomatically interpolated to 512×512 by the scanner. For T₂quantification using Eq. (7), T₁=1.2 s, typical for cartilage at 3T, andthe prescribed flip angle of α=25° were used. Eq. (7) has lowsensitivity to T₁, so small errors in the T₁ assumption should not causelarge errors in estimated T₂. Furthermore, T₁ in knee cartilage does notchange much whereas T₂ is sensitive to degeneration. For comparison, aT₂ map using a full numerical fit was produced from the DESS scan aswell. This involved computing a dictionary of signal ratios for T₂ranging between 10-100 ms (with a step size of 1 ms) using large EPGmatrices (with 6 states) and then choosing the dictionary entry thatbest fitted each pixel. The total scan time was 6:26 min for FSE(acquiring 6 slices) and 5:11 min for DESS (acquiring 36 slices). Inslices where both scans displayed clear anterior and posterior regionsof the femoral cartilage, the resulting T₂ maps in those two regionswere divided into deep and superficial regions, resulting in 4 regionsper slice. This was possible in 10 slices, resulting in a total of 40regions. The root-mean-squared (RMS) difference between the results fromEq. (7) and the FSE results was computed and compared to the differencebetween the results from Eq. (1) and the reference scans. The T₂ mapswere computed in Matlab (The MathWorks, Natick, Mass.) on a Linuxterminal with 128 GB of RAM and 32 2.6 GHz CPUs.

3.3 Patient Scans

As an initial demonstration of the value of Eq. (7) for routinediagnostic imaging, we selected data from three patients from a largergroup scanned with DESS in a clinical setting (spoiler moment=31.32mT/m·ms, spoiler duration=3.4 ms, α=20°, TR=20.4 ms, TE=6.4 ms, FOV=16cm×16 cm, acquisition matrix 416×512, 80 slices with thickness of 1.6 mminterpolated to 0.8 mm (160 slices), 2× parallel imaging in the phaseencode direction, scan time=5:00). T₂ maps were constructedautomatically at the scanner. A proton-density (PD) weighted scan wasacquired for comparison.

4. Results

4.1 Phantom Scans

The results from the phantom scans are shown in FIG. 3. The measuredreference T₂ values were T₂₋₁=62 ms, T₂₋₂=44 ms, and T₂₋₃=31 ms for thethree phantoms. Eq. (1) underestimates T₂ compared to the reference FSEestimate, and this becomes more pronounced for smaller flip angles. Theresults are shown from estimating T₂ from DESS using Eq. (1) (dashed)and Eq. (7) (solid) as well as from reference FSE scans (dotted). Forall phantoms, Eq. (7) clearly agrees better with the reference valuethan Eq. (1), and for flip angles of 20° or higher, Eq. (7) agrees verywell with the reference scans, but gives a substantial overestimate inthe case of a large T₂ and 10° or 15° flip angles. However, the error ismuch smaller than the error from Eq. (1).

4.2 Volunteer Scans

FIGS. 4a-h show the T₂ estimates from the DESS scan, using Eqs. (1) and(7), as well as the results from the reference FSE scans and a fullnumerical fit. The maps from Eq. (7) are more visually similar to thereference scans than those of Eq. (1) (FIGS. 4a-c ). The RMS differencebetween the estimates of Eq. (7) and FSE was 3.7 ms, while for Eq. (1)it was 9.6 ms. Plotting the data points from Eq. (7) against the FSEdata points and drawing a best fit line that crossed the origin yieldeda slope of 0.96, while for Eq. (1) it was 0.77 (FIG. 4 f). Thisdemonstrates that Eq. (7) gave T₂ estimates that better agreed with FSEestimates than Eq. (1). The full numerical fit gives very similarresults to those obtained by Eq. (7), as shown both by its T₂ map and bya difference map (scaled by 10), while taking around 60× longer toproduce, with an average processing time of 577 s per DESS data set(36×512×512 data points) compared to 9 s for Eq. (7).

FIG. 4a shows the first echo (S₁) from a sample sagittal DESS scan. FIG.4b shows the second echo (S₂) from the same DESS scan. The windowinglevel is not the same as in FIG. 4a , in order to show the cartilage.FIG. 4c shows a sample T₂ map from FSE scans of the subject in FIGS.4a-b . FIG. 4d shows a T₂ map of articular cartilage from the DESS scanin panels a-b using Eq. (1). FIG. 4e shows a T₂ map using Eq. (7). FIG.4f shows a T₂ map using a full numerical fit. The map looks very similarto the one in FIG. 4e , but took about 60× longer to produce. FIG. 4gshows a map showing the absolute difference between the maps in FIGS. 4eand f , multiplied by 10. The difference in the cartilage is small,mostly 1 ms or less. Zero difference appears as transparent. FIG. 4hshows the T₂ estimates from 4 regions of femoral cartilage in a total of10 slices from 3 subjects. The trend line for Eq. (7) (green) betteragrees with the FSE scans.

4.3 Patient Scans

FIGS. 5a-d show the scan of a patient with a history of knee injuriesand osteochondral defects, visible on the DESS images. Fibrocartilageformation is present in the damaged hyaline cartilage region and in thesurrounding subchondral bone. The corresponding region in the DESS T₂map shows a focal decrease in T₂ values. FIG. 5a shows the first echo ofa DESS scan of a patient with a chondral lesion (white arrow). FIG. 5bshows the second echo of the DESS scan. FIGS. 5a-b both show cartilagesignal heterogeneity in the central femoral condyle (note the differentwindowing settings). FIG. 5c shows the T₂ maps resulting from applyingEq. (7) to the data in FIGS. 5a-b . FIG. 5d shows a PD weighted scan,acquired for reference, and shows low signal in the lesion.

In another patient (not shown), the availability of the DESS T₂ map madeit possible to visualize an oblique meniscal tear in the anterior hornof the medial meniscus, which was challenging to delineate with only themorphological images. In a third patient (not shown), a T₂ map acquiredwith DESS indicated tendinopathy where the patellar tendon had increasedT₂ values at its insertion site, which was also relatively challengingto visualize with only the morphological DESS images.

5. Discussion

The DESS sequence allows 3D estimation of T₂ in cartilage with high SNRefficiency. The signal expressions are complicated, often necessitatingimprecise simplifications or time-consuming numerical modeling of thesignals. This study explores a linear relationship between the two DESSsignals that can be used for accurately estimating T₂ from a singlescan. Some embodiments have focused on applying the method in cartilage.Other embodiments could be used in various other anatomies. This couldinclude T₂ mapping in breast cancer patients, which has been impracticaldue to the time consuming nature of spin-echo-based T₂ measurements.

The scan time for DESS used in this study was about 5 minutes. Thepatient scans were acquired at a higher resolution than the volunteerscans, but using parallel imaging, the scan time remained at 5 minutesat a cost in SNR. Scan time could be reduced by using lower resolution,higher readout bandwidth, fewer slices, or shorter RF pulses. However,in our experience, a 5 minute scan time is acceptable, especiallyconsidering that these 3D scans provide both morphological andquantitative information. Potentially, DESS could replace sagittal FSEscans that are widely used in many current protocols, but this needs tobe studied further.

As mentioned, using the linear approximation of Eq. (7) for estimatingT₂ from a single DESS scan requires an assumed T₁ value. This should notlead to large estimation errors, since the method has little sensitivityto T₁, which is also not expected to change much in cartilage. Forexample, using the same scan parameters as for the middle curve in FIG.2a , with the same assumption of T₁=1.2 s, yields T₂ estimates of 38.8ms and 41.8 ms (−3.0% and +4.5%) when the real T₁ is 0.96 s and 1.44 s(i.e., the actual T₁ is 20% lower or higher than the assumed value),respectively. Both values are relatively close to the actual value of40.0 ms. A correct assumption of T₁=1.2 s yields T₂=40.6 ms (+1.5%), aslight overestimate due to the approximation made. Similarly,sensitivity to B₁ results in estimates of 36.9 ms and 43.1 ms (−7.75%and +7.75%) when the actual flip angle α is 20° and 30° (i.e., 20% loweror higher than the assumed value of 25°). The B₁ sensitivity could bemitigated by acquiring a B₁ map along with the DESS scan. Thesensitivity of the T₂ estimate to errors in the T₁ and a assumptions isfurther demonstrated in FIG. 6. It should also be noted that assumptionsof T₁ and flip angle will always be needed when estimating T₂ from asingle DESS scan, regardless of the model used. FIG. 6 shows thesensitivity of the T₂ estimation method in Eq. (7) to errors in theassumption of T₁ and B₁. The true T₂ is 40 ms. The flip angle α and T₁are assumed to be 25° and 1.2 s, but actually vary from these values by+/−20%. Other scan parameters were the same as in FIG. 2a . The blackcircle shows the point where the assumptions of a and T₁ are correct.

An embodiment can also be used in combination with other methods forquantitative DESS imaging. An embodiment could estimate other parameterssuch as T₁ or flip angle. Another embodiment would drive from Eq. (6):

$\begin{matrix}{\frac{S_{2H}S_{1L}}{S_{1H}S_{2L}} = {e^{{- {({{TR} - \frac{\tau}{3}})}}\Delta\; k^{2}D}\frac{1 + {e^{- \frac{TR}{T\; 1}}\left( {e^{{- {TR}}\;\Delta\; k^{2}D} - {\cos\mspace{11mu}\alpha}} \right)} - {e^{- \frac{2{TR}}{T\; 1}}\cos\mspace{11mu} a\mspace{11mu} e^{{- {TR}}\;\Delta\; k^{2}D}}}{1 + {e^{- \frac{TR}{T\; 1}}\left( {1 - {\cos\mspace{11mu}\alpha\mspace{11mu} e^{{- {TR}}\;\Delta\; k^{2}D}}} \right)} - {e^{- \frac{2{TR}}{T\; 1}}\cos\mspace{11mu} a\mspace{11mu} e^{{- {TR}}\;\Delta\; k^{2}D}}}}} & (16)\end{matrix}$

This relationship is independent of T₂, and largely insensitive to T₁.

In conclusion, an embodiment provides a simplified expression for theratio between the DESS signals, providing good T₂ estimation. Theexpression disregards higher-order echo pathways, an assumption we haveshown to be valid due to both decay and cancellation of such pathways.The expression takes into account signal dependency to T₁ and flipangle, and we have shown that the method works well by assuming a T₁typical for the tissue and using the prescribed flip angle incalculations.

6. Algorithm

To facilitate understanding, FIG. 7 is a high level flow chart of anembodiment of the invention. A DESS sequence is applied to a volume ofan object by a magnetic resonance imaging (MRI) system (step 704).Signals S₁ and S₂ are acquired by the MRI system from the volume (step708). Signals S₁ and S₂ are used to provide a T₂ map for a plurality ofslices of the volume, comprising determining repetition time (TR), echotime (TE), flip angle α, and an estimate of the longitudinal relaxationtime (T₁), and wherein the DESS scan has a spoiler gradient with anamplitude G and a duration τ and ignoring echo pathways having spentmore than two repetition times in the transverse plane (step 712). Ifthe diffusivity of the tissue D is known, then Eq. (6) can be used alongwith the spoiler gradient amplitude G and duration τ. A T₂ map isdisplayed (step 716).

In one example, Eq. (7) may be used to provide the T₂ map from signalsS₁ and S₂ with known values of TR, TE, α, and longitudinal relaxationtime (T₁). The use of Eq. (7) allows echo pathways that have spent morethan two repetition times in the transverse plane to be ignored. Inaddition, the use of Eq. (7) allows the T₂ map to be generated, whileignoring diffusivity, and estimating T₁. In another example, diffusivityis measured and Eq. (6) is used to provide the T2 map from signals S₁and S₂ with known values of TR, TE, α, and an estimate of T₁.

FIG. 8 is a schematic top view of a magnetic resonance imaging (MRI)system 800 that may be used in an embodiment of the invention. The MRIsystem 800 comprises a magnet system 804, a patient transport table 808connected to the magnet system, and a controller 812 controllablyconnected to the magnet system. In one example, a patient would lie onthe patient transport table 808 and the magnet system 804 would passaround the patient. The controller 812 would control magnetic fields andradio frequency (RF) signals provided by the magnet system 804 and wouldreceive signals from detectors in the magnet system 804.

FIG. 9 is a high-level block diagram showing a computer system 900,which may be used to provide the controller 812. The computer system mayhave many physical forms ranging from an integrated circuit, a printedcircuit board, and a small handheld device up to a computer. Thecomputer system 900 includes one or more processors 902, and further caninclude an electronic display device 904, a main memory 906 (e.g.,random access memory (RAM)), storage device 908 (e.g., hard disk drive),removable storage device 910 (e.g., optical disk drive), user interfacedevices 912 (e.g., keyboards, touch screens, keypads, mice or otherpointing devices, etc.), and a communication interface 914 (e.g.,wireless network interface). The communication interface 914 allowssoftware and data to be transferred between the computer system 900 andexternal devices via a link. The system may also include acommunications infrastructure 916 (e.g., a communications bus,cross-over bar, or network) to which the aforementioned devices/modulesare connected.

Information transferred via communications interface 914 may be in theform of signals such as electronic, electromagnetic, optical, or othersignals capable of being received by communications interface 914, via acommunication link that carries signals and may be implemented usingwire or cable, fiber optics, a phone line, a cellular phone link, aradio frequency link, and/or other communication channels. With such acommunications interface, it is contemplated that the one or moreprocessors 902 might receive information from a network, or might outputinformation to the network in the course of performing theabove-described method steps. Furthermore, method embodiments of thepresent invention may execute solely upon the processors or may executeover a network, such as the Internet, in conjunction with remoteprocessors that shares a portion of the processing.

The term “non-transient computer readable medium” is used generally torefer to media such as main memory, secondary memory, removable storage,and storage devices, such as hard disks, flash memory, disk drivememory, CD-ROM and other forms of persistent memory and shall not beconstrued to cover transitory subject matter, such as carrier waves orsignals. Examples of computer code include machine code, such asproduced by a compiler, and files containing higher level code that areexecuted by a computer using an interpreter. Computer readable media mayalso be computer code transmitted by a computer data signal embodied ina carrier wave and representing a sequence of instructions that areexecutable by a processor.

In an example, a patient may be placed on patient transport table 808 ofthe MRI system 800. The controller 812 causes the magnet system 804 toprovide a DESS sequence for a volume of the patient (step 704). In thisexample, the volume is a knee of the patient. Signals S₁ and S₂ areacquired by the MRI system 800 (step 708). The signals S₁ and S₂ areused to provide a T₂ map for a plurality of slices of the knee bydetermining repetition time (TR), echo time (TE), flip angle α,longitudinal relaxation time T₁, and where the DESS scan has a spoilergradient with an amplitude G and a duration τ and ignoring echo pathwayshaving spent more than two repetition times in the transverse plane(step 712).

In this embodiment, S₁ and S₂ are used to provide a T₂ map using afunction

${T_{2} = \frac{{- 2}\left( {{TR} - {TE}} \right)}{\log\left\lbrack {\left( \frac{S_{2}}{S_{1}\sin^{2}\frac{\alpha}{2}} \right){f\left( {\alpha,{TR},{T\; 1}} \right)}} \right\rbrack}},$where ƒ(α, TR, T1) is a function that is not proportional to

$\sin^{2}{\frac{\alpha}{2}.}$

More preferably, in this embodiment

${f\left( {\alpha,{TR},{T\; 1}} \right)} = {\frac{\left( {1 - {\left( {\cos\mspace{11mu}\alpha} \right)e^{\frac{- {TR}}{T\; 1}}}} \right)}{\left( {1 + e^{- \frac{TR}{T\; 1}}} \right)}.}$

It should be noted that the equation used in this embodiment, is thesame as Eq. (7), when solving for T₂. Therefore, using Eq. (7), in thespecification and claims would be the same as using the equation in thisembodiment. As mentioned above, Eq. (7) may be used to provide the T₂map from signals S₁ and S₂ with known values of TR, TE, α, and anestimate of the relaxation time (T₁). The use of Eq. (7) allows echopathways that have spent more than two repetition times in thetransverse plane to be ignored. In addition, the use of Eq. (7) allowsthe T₂ map to be generated, while ignoring diffusivity, and estimitingT₁.

In this example, the controller 812 may be used to generate and thendisplay an image on the display 904 (step 716). In this example, aplurality of T₂ map images are displayed, where each T₂ image is of aslice of the volume, so that the plurality of T₂ map images from aplurality of slices of the knee to provide a 3D view of the knee.

MRI systems signals measure magnetization that is perpendicular to themain magnetic field. Perpendicular signals are said to be in thetransverse plane. The signals measured in steady-state sequences such asDESS are composed of magnetization components that have evolved overmany repetition times (TRs). During some TRs, the magnetization isparallel to the main magnetic field, and during others, it isperpendicular to the field (in the transverse plane). In thespecification and claims, ignoring echo pathways with more than tworepetition times in the transverse plane means that it is assumed thatthe measured signal does not have magnetization components that havespent more than two repetition times in the transverse plane.

While this invention has been described in terms of several preferredembodiments, there are alterations, permutations, modifications andvarious substitute equivalents, which fall within the scope of thisinvention. It should also be noted that there are many alternative waysof implementing the methods and apparatuses of the present invention. Itis therefore intended that the following appended claims be interpretedas including all such alterations, permutations, modifications, andvarious substitute equivalents as fall within the true spirit and scopeof the present invention.

What is claimed is:
 1. A method for providing an estimated 3D T₂ map formagnetic resonance imaging using a Double-Echo Steady-State (DESS)sequence for a volume of an object in a magnetic resonance imaging (MRI)system, comprising: providing a DESS scan of the volume by the MRIsystem, wherein the DESS scan has a spoiler gradient with an amplitude Gand a duration τ; acquiring signals S₁ and S₂ by the MRI system; andproviding a T₂ map for a plurality of slices of the volume based onsignals S₁ and S₂, the step of providing the T₂ map further comprisingdetermining repetition time (TR), echo time (TE), flip angle α, and anestimate of the longitudinal relaxation time (T₁), and ignoring echopathways having spent more than two repetition times in the transverseplane, wherein the step of providing a T₂ map determines T₂ using afunction${T_{2} = \frac{{- 2}\left( {{TR} - {TE}} \right)}{\log\left\lbrack {\left( \frac{S_{2}}{S_{1}\sin^{2}\frac{\alpha}{2}} \right){f\left( {\alpha,{TR},{T\; 1}} \right)}} \right\rbrack}},$wherein f(α, TR, T1) is a function that is not proportional to$\sin^{2}\frac{\alpha}{2}$ and TR is a repetition time, TE is an echotime, and α is a flip angle; and displaying the T₂ map.
 2. The method,as recited in claim 1, wherein${f\left( {\alpha,{TR},{T\; 1}} \right)} = {\frac{\left( {1 - {\left( {\cos\mspace{11mu}\alpha} \right)e^{\frac{- {TR}}{T\; 1}}}} \right)}{\left( {1 + e^{- \frac{TR}{T\; 1}}} \right)}.}$3. A method for providing an estimated 3D T₂ map for magnetic resonanceimaging using a Double-Echo Steady-State (DESS) sequence for a volume ofan object in a magnetic resonance imaging (MRI) system, comprising:providing a DESS scan of the volume by the MRI system; acquiring signalsS₁ and S₂ by the MRI system; providing a T₂ map for a plurality ofslices of the volume based on signals S₁ and S₂, using the equation${T_{2} = \frac{{- 2}\left( {{TR} - {TE}} \right)}{\log\left\lbrack {\left( \frac{S_{2}}{S_{1}\sin^{2}\frac{\alpha}{2}} \right){f\left( {\alpha,{TR},{T\; 1}} \right)}} \right\rbrack}},$wherein f(α, TR, T1) is a function that is not proportional to$\sin^{2}\frac{\alpha}{2}$ and TR is a repetition time, TE is an echotime, and a is a flip angle; and displaying the T₂ map.
 4. The method,as recited in claim 3, wherein${f\left( {\alpha,{TR},{T\; 1}} \right)} = {\frac{\left( {1 - {\left( {\cos\mspace{11mu}\alpha} \right)e^{\frac{- {TR}}{T\; 1}}}} \right)}{\left( {1 + e^{- \frac{TR}{T\; 1}}} \right)}.}$